Integer partitions are finite multisets of natural numbers, which ripple with interesting and useful set-theoretic properties. We denote a generic partition \(\lambda =(\lambda _1, \lambda _2, \lambda _3, \dots , \lambda _r), \lambda _1\ge \lambda _2 \ge \dots \ge \lambda _r\ge 1\) , with \(\emptyset \)  denoting the empty partition. There are a number of natural statistics on partitions that are of interest in number theory and combinatorics, such as the size \(|\lambda |=\sum _{1\le i \le r}\lambda _i\)  (sum of parts) and the related partition function \(p(n)\)  that counts the number of size-\(n\)  partitions, \(n\ge 0\) ; the length \(\ell (\lambda )\)  (number of parts); the largest part \(\text{lg}(\lambda )=\lambda _1\)  and smallest part \(\text{sm}(\lambda )=\lambda _r\)  of the partition; the multiplicity \(m_i=m_i(\lambda )\ge 0\)  (frequency) of the part \(i\ge 1\)  in \(\lambda \) ; along with Dyson's rank, the Andrews-Garvan crank, and other useful maps from the set of partitions \(\mathcal {P}\)  to \(\mathbb {N}\)  (see e.g. [1]}).
